If your rig's recent you can create a first primary bootable 8GB or 16GB FAT32 partition and boot from it (you only need 5GB for windows' setup files, the rest leave it for just in case and maybe some drivers) on the USB drive, create a secondary partition and format as NTFS like always.
BACKUP FIRST (you have to empty the USB external drive)

Doesn't work on all computers, unless recent (uefi boot) but it's the fastest way to get it done,

Just format a USB thumbdrive FAT32 and extract the iso to it and you're done, bootable on UEFI or Legacy BIOS systems.

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Well, yes, but the OP's stipulation, as stated in the title, is for a clean install to be performed without a DVD or a USB stick. In the post it's stated that they don't have a USB stick that's bigger than 4GB (which I take to mean that smaller ones are lying around) but OP does have an external hard drive with stuff already on it. Most UEFI implementations won't let you boot directly off an NTFS drive (patents, I assume), hence the stuff with a flash drive to be able to get the UEFI to read from that hard drive. This way, nothing gets lost on the external drive (well, unless a mistake is made during setup). The only thing I don't know is if OP is doing an EFI install.

Well, yes, but the OP's stipulation, as stated in the title, is for a clean install to be performed without a DVD or a USB stick. In the post it's stated that they don't have a USB stick that's bigger than 4GB (which I take to mean that smaller ones are lying around) but OP does have an external hard drive with stuff already on it. Most UEFI implementations won't let you boot directly off an NTFS drive (patents, I assume), hence the stuff with a flash drive to be able to get the UEFI to read from that hard drive. This way, nothing gets lost on the external drive (well, unless a mistake is made during setup). The only thing I don't know is if OP is doing an EFI install.

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Yeah the 4GB limit i did not take in consideration.

But he simply could do a reset (when already on 10) or open the iso on the running desktop and upgrade with the option to take no files or settings to the new install (can't provide the correct text or sshot atm, sorry).

I've made an empty iso (all unneeded files deleted, result = 41MB), when you want you can extract the empty iso (based on 14393 Pro x64 en-US) to the FAT32 formatted usb thumbdrive and ad your own boot.wim and install.esd and it will fit on a 4GB usb drive. If you are interested in the empty iso, send me a pm and i'll provide you a downloadlink. If you don't trust it (OR want the iso in another language) you can recreate your own empty iso by deleting all the files from a clean iso with the exeption of the files shown in my empty iso.

I've made an empty iso (all unneeded files deleted, result = 41MB), when you want you can extract the empty iso (based on 14393 Pro x64 en-US) to the FAT32 formatted usb thumbdrive and ad your own boot.wim and install.esd and it will fit on a 4GB usb drive. If you are interested in the empty iso, send me a pm and i'll provide you a downloadlink. If you don't trust it (OR want the iso in another language) you can recreate your own empty iso by deleting all the files from a clean iso with the exeption of the files shown in my empty iso.